Average Error: 2.8 → 1.3
Time: 8.1s
Precision: binary64
\[x \ge 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{1}{\sqrt{\pi}} \cdot \left(\left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}\right) \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\]
\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)
↓
\frac{1}{\sqrt{\pi}} \cdot \left(\left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}\right) \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)
double code(double x) {
	return ((double) (((double) (((double) (1.0 / ((double) sqrt(((double) M_PI))))) * ((double) exp(((double) (((double) fabs(x)) * ((double) fabs(x)))))))) * ((double) (((double) (((double) (((double) (1.0 / ((double) fabs(x)))) + ((double) (((double) (1.0 / 2.0)) * ((double) (((double) (((double) (1.0 / ((double) fabs(x)))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))))))) + ((double) (((double) (3.0 / 4.0)) * ((double) (((double) (((double) (((double) (((double) (1.0 / ((double) fabs(x)))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))))))) + ((double) (((double) (15.0 / 8.0)) * ((double) (((double) (((double) (((double) (((double) (((double) (((double) (1.0 / ((double) fabs(x)))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x))))))))))));
}
↓
double code(double x) {
	return ((double) (((double) (1.0 / ((double) sqrt(((double) M_PI))))) * ((double) (((double) (((double) pow(((double) exp(((double) fabs(x)))), ((double) (((double) fabs(x)) / 2.0)))) * ((double) pow(((double) exp(((double) fabs(x)))), ((double) (((double) fabs(x)) / 2.0)))))) * ((double) (((double) (1.0 / ((double) fabs(x)))) + ((double) (((double) (((double) (1.0 / 2.0)) * ((double) pow(((double) (1.0 / ((double) fabs(x)))), 3.0)))) + ((double) (((double) (((double) (3.0 / 4.0)) * ((double) pow(((double) (1.0 / ((double) fabs(x)))), 5.0)))) + ((double) (((double) (15.0 / 8.0)) * ((double) (1.0 / ((double) pow(((double) fabs(x)), 7.0))))))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.8

    \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

  2. Simplified1.4

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{7}\right)\right)\right)\right)}\]

  3. Taylor expanded around 0 1.2

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{7}}}\right)\right)\right)\right)\]
  4. Using strategy rm

  5. Applied sqr-pow1.3

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\]

  6. Final simplification1.3

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(\left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}\right) \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3} + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))